What is the natural base of e?

Answer 1

'e' is an irrational number ... it can't be written completely using digits, fractions,

or decimals, so it has to be written as a symbol, just like 'pi'.

Here's the beginning of it: 2.718281828459045 . . .

'e' is a number that keeps popping up, when you do the math for any kind of process

where the speed depends on how far it still has to go to reach the finish line.

Famous example . . . Picture this: The width of your bedroom is 'W'. One morning,

you stand with your back to the wall at one side of your bedroom, and when your

Mom says 'Go', you start walking across the room to the other side, like this:

Every second, you walk 1/2 of the distance you have left.

For one thing, you'll never reach the other side, because you can only cover

1/2 of the remaining distance in the next second. But that's another story.

After 'T' seconds, the distance you have left to go is: W e-0.69T

and there's an example of how that number 'e' keeps coming up.

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My improvement to the previous answer is not an improvement to the actuall value of e previously mentioned, but another description of what e, or the value actually means: as to why 2.71828 and not some other value. I try to explain why it is then called "natural".

e or epsilon is about 2.71828 e is a very important number like PI = 3.14...

This is called a natural number because it occurs in nature alot like the Fibonacci series or numbers. e can be used to calculate natural growths and decays. For example an object that is cooling down, the temperature can be calculated where the formula for that has e in it. A series for e has been developed which has great power in finding not only powers of e but powers and roots of any value since series for those values have not been developed (such as a series for powers of 10).

The definition of e is based upon (1/1 + 1/n)^n where n is very high. Quickly though it reaches 2.718 even when n is lower. The value of this expression approaches the value of a constant we will just call e. If you have infinitely compounded growth during one period of time, the initial value will grow to about 2.718 times bigger for that entire period, and not some enormous value like some might think initially. The reason is that the compounded growth value to be added in gets very small when the number of intervals (n in the formula) of compounded growth is very high.

Because of the natural nature of e, it was chosen as a base for logarithms. When e is used, the logarithm is then often called a "natural logarithm".

In short the derivative of (ln x) is (1/x), ie. the reciprocal of that x, and then a reciprocal series (of terms to be calculated) was developed to find the natural logarithm of a number. This series for the natural logarithm of a number is very powerful since series to calculate other logarithms with different bases have basically never been developed yet. In essence, logarithms of other bases are calculated with the natural logarithm series and then the resulting value is converted or adjusted (usually with a multiplication by a value) to what is should actually be.

Another interesting fact about e, is that the derivative of e is e or itself. Likewise, the derivative of any other power of e is that same value. In short, this means that on a curve on a graph, the steepness or slope of that curve at that point is equal to that current value of e, that is, itself.